(*
  DaoMath Advanced Duality Theorems
*)

Require Import DualPair.
Require Import Quantum.

(** * Pontryagin 对偶 *)

(** 对偶的对偶同构于原空间 *)
Theorem pontryagin_duality :
  forall (X : Type) (dual_X : Type),
  (* X** ≅ X *)
  exists (phi : X -> dual_X -> R),
  forall x : X,
  exists f : dual_X -> R,
  forall g : dual_X -> R,
  f = g <-> (forall y, phi x y = g y).
Proof.
Admitted.

(** * 量子-经典对偶 *)

Theorem quantum_classical_duality :
  forall (psi : QuantumState),
  is_normalized psi ->
  exists (prob_dist : R -> R),
  forall x : R,
  prob_dist x >= 0 /\
  (* ∫ prob_dist dx = 1 *)
  True. (* 需要积分理论 *)
Proof.
Admitted.
